This paper deal with the study of local convergence of fourth and fifth order iterative method for solving nonlinear equations in Banach spaces. Only the premise that the first order Fréchet derivative fulfills the Lipschitz continuity condition is needed. Under these conditions, a convergence theorem is established to study the existence and uniqueness regions for the solution for each method. The efficacy of our convergence study is shown solving various numerical examples as a nonlinear integral equation and calculating the radius of the convergence balls. We compare the radii of convergence balls and observe that by our approach, we get much larger balls as existing ones. In addition, we also include the real and complex dynamic study of one of the methods applied to a generic polynomial of order two.

中文翻译：

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Banach空间中四阶和五阶参数族迭代方法的局部收敛

本文研究了求解巴拿赫空间中非线性方程的四阶和五阶迭代法的局部收敛性。只需要一阶 Fréchet 导数满足 Lipschitz 连续性条件的前提。在这些条件下，建立收敛定理来研究每种方法解的存在唯一性区域。我们的收敛研究的功效显示为将各种数值示例求解为非线性积分方程并计算收敛球的半径。我们比较了收敛球的半径，并观察到通过我们的方法，我们得到了比现有球大得多的球。此外，我们还包括对应用于二阶泛型多项式的方法之一的实数和复数动态研究。